Decomposition of monotone functions
Robert O’Brien
Source abstract
A monotone surjective map F F is continuous and defines a spectral family and unitary group on L 2 L^{2} . We use the H 0 \mathcal {H}_{0} splitting theorem (Int. J. Dyn. Syst. Differ. Equ. 10 (2020), no. 4, 358–372) to extend the Lebesgue decomposition F = F s + F a F \!=\! F_{s} \!+\! F_{a} to the singular component of F F , F s = F m + F w s F_{s} \!=\! F_{m} + F_{w_{s}} . The components ( recurrent : F m F_{m} , and stable : F w s F_{w_{s}} ) are unique up to additive constants and are characterized by their Fourier–Stieltjes transforms. Several examples are included. Section 7.2 shows that each Riesz product has a nontrivial recurrent component.
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